Evolution of the Hydrogen Atom

The hydrogen atom remained the same and will remain the same, not like
the human race. As for the way we look it, there have been some
evolutions.

The hydrogen atom played its unique role as a physical model for bound states in
quantum mechanics. Niels Bohr thought both the proton and electrons were point
particles, and the proton was sitting at the center of the absolute frame.

In quantum mechanics, the picture of standing waves replaces the electron orbit,
creating discrete energy levels. Since we cannot accelerate the hydrogen
atom, there does not seem to be any urgent need for studying how the
orbit would appear to observers in different Lorentz frames.

Yet, the Lamb shift played the key role in the development of quantum electrodynamics,
which is a Lorentz-covariant theory. This aspect is well known. what is
not well known is the fact that the Lamb shift calculation still needs localized
hydrogen wave functions which are constructed in Bohr's absolute frame.
Do you know how the wave functions appear to moving observers?

In his book entitled "Speakable and Unspeakable in Quantum Mechanics"
(page71), J. S. Bell talks about how the electron orbit would appear if the
hydrogen atom moves and came up with this figure
this drawing. Yes, this Lorentz-deformed orbit is speakable,
but it is not observable, because the hydrogen atom cannot be accelerated.

The story is quite different if the electron is removed from the hydrogen atom.
Indeed, the proton speed can reach the speed of light times 0.99999-- ( I do not
know how many nines). Then what aspect of the hydrogen orbit can we study from
those accelerated protons? What physics does the proton share with the hydrogen
atom?

The proton is one of the hadrons which are quantum bound states of quarks.
The hydrogen atom is a bound state.

Like the hydrogen atom, hadrons have ground and excited states. Their
mass spectra are called Regge trajectories. Many people think God created
those Regge poles on the trajectories, but
Feynman pointed out
they represent degeneracies of the three-dimensional harmonic oscillators.
This aspect is not well known in spite of Feynman's presage.

The question is whether we can entertain Bell's picture of Lorentz-deformed
wave function using harmonic oscillator wave functions. Three-dimensional
oscillator wave functions are separable in the Cartesian coordinate system,
and thus the longitudinal coordinate can be treated separately.

In 1969, Feynman came up with the idea of partons. If the proton moves
with a velocity close to that of light, it appears as collection of free
partons behaving incoherently with external signals. Partons are routinely
assumed to be quarks in the papers on high-energy physics. The question is
whether these two different physical models are two different manifestations
of one covariant entity. This question has been addressed in this
webpage.

Form factor as a function of the (momentum transfer)2 .

Our next question is how the hadron would look if it gradually picks up.
In order to study the problem, let us consider the scattering of election
and proton. If the electron comes close to the proton, it emits a photon,
which is absorbed by the proton. If the proton is a point particle, we can
calculate the Rutherford formula for the scattering amplitude using one-photon
exchange Feynman diagrams.

But the Hofstadter
experiment of 1955 showed
the deviation from this formula indicating that the proton has a non-point
charge distribution. I first heard about this in 1958 while I was doing
a homework for my QM course at Carnegie Tech. I was a graduating senior
then. Here is the homework problem.

During this collision process, the momentum is also transferred. As the
momentum transfer increases, the deviation from the Rutherford scattering
becomes more prominent. This is consistent with an idea that the charge
of the proton is distributed, causing the deviation from the Rutherford
formula.

Thus, instead of a point in the Feynman diagram, we put a distribution
function. Indeed, this is consistent with the quark model, where the
proton is a bound state of the quarks. The Fourier transform of this
density distribution with respect to the momentum transfer is called
the proton "form factor."

If the proton wave function is Gaussian, its charge distribution
is also Gaussian. If we take the Fourier transform of this Gsussian
function, the form factor should decrease like Gaussian, but this
is not what is happening in the world. The decrease is much slower.
The form factor goes through a polynomial decrease. This decrease is
known as the polynomial decrease.

What is the cause of this slower decrease? In order to explain this,
let us exploit J.S.Bell's picture of
Lorentz deformation. If we go to the Breit frame where the incoming
momentum and outgoing momentum are in the opposite directions, as shown
below. Then there are papers in the literature leading to the
Lorentz-contracted distribution function like Bell's ellipse.

How does this figure explain the slower cutoff. The answer is the
"coherence" as shown below. The width of the proton distribution
decreases as the wavelength of the incoming photon decreases.